Geometric Probability Distributions

1
Geometric Probability Distributions
2
Introduction

• We have been looking at Binomial Distributions
• A family has 3 children. What is the probability
  they have 2 boys?
• Johnny rolls a die 6 times. What is the
  probability that he will roll a 2 three times?
• Now lets look at these as Geometric
  Distributions
• A family will have kids until they have a boy
• Johnny rolls a die until he rolls a 2

3
What Gives Us a Geometric Setting?

• There are 4 key elements necessary to have a
  geometric setting. They are as follows
• Each observation is either a Success or
  Failure
• The n observations are all independent.
• The probability of success, p, is the same for
  each observation
• The variable of interest is the number of
  trials required to obtain the first success

Lets look at these elements in the context of a
scenario.
4
What Gives Us a Geometric Setting?

• Scientists have randomly selected 10 rats known
  to have the flu and will inoculate them with a
  vaccine until one is cured.
• Each observation is either a Success or
  Failure
• In this case, the rats will either be Cured or
  Not Cured
• The variable of interest is the number of trials
  required to obtain the first success
• We are looking to find probabilities that the 1st
  or the 2nd or the 3rd rat (etc.) gets cured
• The n observations are all independent.
• The results of the vaccine in one rat do not
  effect the results for the next rat
• The probability of success, p, is the same for
  each observation
• The probability of success for each rat is .65

5
Examining the Lingo

• Geometric probabilities
• Probability of Xn is the probability that the
  1st success, X, is on the nth trial
• n number of trials
• p Probability of Success on any ONE observation
• Remember you must 1st have a geometric setting
  (meet the 4 criteria) before you can calculate a
  geometric probability
• Geometric Probability Distributions are
  technically infinite but as you increase trials,
  the probability will get closer and closer to 0

6
Finding Geometric Probabilities

• Below is the formula for calculating a geometric
  probability. To display a distribution, you just
  continue to calculate the probabilities
• Lets look at the formula

This is basically failurefailurefailuresucces
s. Where you have n-1 failures before you
succeed
Want to do this with a calculator function
geometpdf(p,x) where x is the trial you are
looking for and p is the probability of success
7
Lets Practice Using the Formula and the
Calculator

• Allen Iverson is a career 73 free throw shooter.
  Find the probabilities below.

1. Find the probability that Allen makes his
first shot.
geometpdf(.73,1) .73
2. Find the probability that it takes Allen two
shots to make one.
geometpdf(.73,2) .1971
3. Find the probability that Allen shoots five
shots before he makes one.
geometpdf(.73,6) .0010
4. Find the probability that Allen makes takes no
more than four shots to make one.
geometpdf(.73,1) geometpdf(.73,2)
geometpdf(.73,3) geometpdf(.73,4)
8
Cumulative Function

• We can use the cumulative function for geometrics
  as well
• They give you the probability of n trials or less
• geometcdf(p,x)
• This gives you the sum of the probabilities from
  1 to X

Find the probability that Allen makes takes no
more than four shots to make one.
geometcdf(.73,4) .9947
You do MORE THAN probabilities by subtracting the
cdf from 1, just like in the binomial setting.
9
Get Some Answers!!!

• Find the probabilities below
• Probability Allen takes more than 4 shots to make
  his first.
• Probability it takes Allen more than 6 shots to
  make his first.
• Probability Allen makes his first shot in on his
  8th try.

1 geometcdf(.73,4) .0053
1 geometcdf(.73,6) .0004
geometpdf(.73,8) .00008
10
The Pesky Formula

• Below is the by hand formula used for MORE than
  probabilities

Roll a die until a 3 is observed. The
probability that it takes more than 6 rolls to
observe a 3 is
P(Xgt6) (1-1/6)6 (5/6)6 .335
11
Geometric Mean Spread
Geometric Mean
Geometric Variance
Geometric Standard Deviation
12
Homework
Read Pages 472,73 on Simulating Geometric
Experiments
Do Problem s 41 - 51